Spin Valve Effect in ZigZag Graphene Nanoribbons by Defect Engineering
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Spin Valve Effect in ZigZag Graphene Nanoribbons by Defect Engineering
Sankaran Lakshmi , Stephan Roche , and Gianaurelio Cuniberti Institute for Materials Science and Max Bergmann Center of Biomaterials,Dresden University of Technology, D-01062 Dresden, Germany CEA, Institute for Nanoscience and Cryogenics, INAC/SP2M/ L sim ,17 rue des Martyrs, 38054 Grenoble Cedex 9, France We report on the possibility for a spin valve effect driven by edge defect engineering of zigzaggraphene nanoribbons. Based on a mean-field spin unrestricted Hubbard model, electronic bandstructures and conductance profiles are derived, using a self-consistent scheme to include gate-induced charge density. The use of an external gate is found to trigger a semiconductor-metaltransition in clean zigzag graphene nanoribbons, whereas it yields a closure of the spin-split bandgapin the presence of Klein edge defects. These features could be exploited to make novel charge andspin based switches and field effect devices.
The recent fabrication of single and few layersgraphene has sparked considerable expectations in thefield of all-carbon nanoelectronics [1, 2]. Although 2Dgraphene is a zero gap semiconductor, the possibility topattern graphene nanoribbons (GNRs) with widths offew tens of nm , has enabled band gap engineering [3]and the development of efficient GNRs-based field ef-fect transistors [4]. However, the further optimization ofGNRs-based devices or the design of novel device func-tionalities demand for more understanding and control ofnot only unavoidable structural disorder and defects, butalso edge geometry and chemical functionalization. Ex-perimental characterization either with Raman studies,scanning tunneling microscopy or high-resolution elec-tronic transmission microscopy has revealed a large spec-trum of topologically different edge profiles, which canexhibit either an armchair or a zigzag symmetry [5], aswell as a zoology of edge imperfections [6]. In particular,individual pending atoms named Klein defects have beenpredicted [7] and more recently observed [8, 9]. In prin-ciple, such edge disorder is unsuitable for keeping goodconduction properties of otherwise clean GNRs. As amatter of fact, several experimental [10] and theoretical[11] reports evidence the enlargement of the transportgap and strong fluctuations of low temperature conduc-tance.A pioneering work by Son et al. [12] has unveiled half-metallic behavior in clean ZGNRs in the presence of atransverse electric field, opening perspectives for the de-sign of spin-dependent switching devices [13]. Severalrecent studies have also reported on the possibility forhalf-metallicity in ZGNRs using chemical doping [14],whereas the application of an external magnetic field wasshown to trigger a transition from parallel to antiparal-lel magnetic edges, resulting in giant magnetoresistancephenomena and efficient spintronics devices [15].In this Letter, we investigate the electronic and con-ductance properties of gated zigzag GNRs (ZGNRs) withand without the presence of Klein edge defects. The en-ergetics of the GNRs is described by a mean-field Hub-bard Hamiltonian which has been shown to well repro- duce density functional theory results in the spin localdensity approximation [16, 17]. The gate-injected chargedensity is also self consistently included in the band struc-tures and transport calculations. For clean zigzag GNRs,the external gate is shown to tune the electronic structurefrom a semiconducting state to a purely metallic state,thus switching on current through the device, whereasin presence of Klein defects, the spin-split gap is closed,resulting in a transition from a pure spin-current to aspin-degenerate charge current, close to the Fermi en-ergy.ZGNRs are very peculiar in the sense that they have amagnetic ground state, with each edge having electronsaligned ferromagnetically with each other, albeit anti-ferromagnetically with respect to the other edge [12].The ground state is entirely dominated by these mag-netic edge states [5], which generate a semiconductingband gap [12]. The introduction of a Klein edge profile(with additional π -electron hopping bonds) in the ZGNR,is known to produce a flat band over the entire Brillouinzone [8], within a tight-binding description. However,with the incorporation of electron correlations, a groundstate with a net magnetization and a spin-split band gapemerges [18, 19]. Kusakabe et al. have theoretically dis-cussed a Klein edged situation by dihydrogenating oneof the edges of the ZGNR, validating the occurence ofa spin-polarized band at the Fermi energy [19]. Veryrecently, these edges were observed for the first time, al-though locally, in graphene using HR-TEM [9].Here, it is our interest to study the effect of an externalgate on the band gaps of these two categories of bipar-tite semiconducting systems, one of which has a anti-ferromagnetic ground state (ZGNR), and the other fer-romagnetic (ZGNR with a Klein edge defect). We modelGNRs using the spin-unrestricted mean-field HubbardHamiltonian, H = − t X h ij i ,σ ( c † iσ c jσ + h.c) + U X i,σ h n i − σ i n iσ (1)where c † iσ , c iσ and n iσ are the creation, annihilation and -505 E - E F [ e V ] -505-0.6 0 0.6k z [ π/ a]-505 E - E F g [ e V ] -0.6 0 0.6k z [ π/ a] -505 E - E F [ e V ] E - E F g [ e V ] (b) (c)(a) (d) Z FIG. 1: (color online) Band structures of the clean ZGNR with (a) V g = 0 (b) V g = 1 . V g = 0 (d) V g = 1 . number operators for an electron of spin σ in the π orbitalof the i th C atom in the ribbon, with t = 2 . U =2 .
75 eV [11]. The unit cell of the ZGNR has width 20 (40atoms) and that of the Klein edged ZGNR has an extraC atom at one edge. The source-drain electrodes, forsimplicity are assumed to consist of the same GNRs. TheFermi energy of the non-gated system ( E F ) is obtainedby integrating the charge density up to half-filling.One common way to tune the device conductance is touse an external metallic gate for monitoring the depletionor accumulation of charges in the conducting channel [4].The presence of a third (gate) terminal is numericallysimulated by shifting the on-site energies by eV g where V g is the gate voltage. The modified charge density distri-bution up to E F at every V g is further self-consistentlycomputed. To do so, the excess charge carriers intro-duced by the gate are obtained as n ( E gF ) − n ( E F ) where n ( E gF ) is the charge density at the Fermi energy in thepresence of the gate and n ( E F ) is that in the absenceof the gate ( n ( E F ) = R E F −∞ ρ ( E ) dE , with ρ ( E ) the den-sity of states). The modified charge density at every V g is then incoporated into the Hamiltonian and solved self-consistently, to obtain the new band-stucture of the gatedGNR, with the associated Fermi level ( E gF ).The coherent transport through the system is then calculated, using the well-known Landauer’s formalism.The retarded Green function of the device is computedas G = ( E − H − Σ L − Σ R ) − where Σ L and Σ R arethe self-energies of the GNR electrodes, calculated asΣ = τ g s τ † where τ contains the device-source/drain in-teractions and g s is the surface Green’s function of theelectrodes, calculated using standard recursive Green’sfunction techniques [11]. The conductance of the de-vice is finally obtained as G D ( E ) = e h T ( E ) where T(E),the transmission probability at energy E is given byTr[Γ L G Γ R G † ], Tr is the Trace and Γ L,R = i (Σ L,R − Σ † L,R ).Electronic band structures of ZGNR with or withouta Klein edge, and as a function of applied external gatescan be seen in Fig 1. First, as observed in Fig 1a, when V g = 0, the ground state is characterized by a bandgap, resulting from its oppositely spin-polarized edges.With the application of a positive gate voltage channel-ing holes into the system, the electron density at theedges (2 / ≤ | k | ≤
1) gradually diminishes, which resultsin a completely non-magnetic, metallic system. A closingof the band gap is seen in Fig 1b for V g = 1 . C ondu c t a n ce ( e / h ) -0.5 0 0.502468 (a)(b) gF E − E [eV]
FIG. 2: Spin-degenerate (up and down) conductance of theZGNR with respect to the incident energy of the electrons (E)scaled by E gF for (a) V g = 0 (b) V g = 1 . by the number of available conduction channels at therelevant Fermi level. At injection energies close to E gF ,a current switch is driven by the gate voltage increase,indicating a transistor type behavior.In contrast, the situation for the ZGNR with one Kleinedge shows some marked differences. First by looking atthe band structure at zero gate voltage (Fig.1c), it isclear that the highest occupied band is made up entirelyof one spin (dotted) and the lowest unoccupied band, ofthe other (solid). The ground state of the system showssome ferromagnetism, with both edges displaying highelectron density of the same (majority) spin. This is ex-plained by the finite sublattice imbalance which favorsthe appearance of midgap states and magnetic proper-ties [16, 17, 20].When a positive gate voltage is applied, the major-ity spin (up) band shifts upward, reducing the electron(up spin) density, as seen in Fig 3b, which eventuallymeets the minority spin (down) density to make the sys-tem completely non-magnetic and metallic. This is alsocharacterized in Fig 3a by the slow reduction of the spin-split gap, which eventually disappears beyond V g = 1 . V g = 1 . V g = 0, the transmission probability of themajority spin close to E gF is very high, as it is the onlyband available in the energy range. As the gate voltageis increased, the system transmits only the majority spinuntil ≈ . S p i n - s p lit g a p ( e V ) -2 -1 0 1 2 V g N o . o f e l ec t r on s (a)(b) FIG. 3: (color online) (a) Variation of the bandgap over theentire Brillouin zone with V g (b) The total number of up (reddashed) and down electrons (black solid) with respect to V g .Dotted lines are marked at V g = 0 and V g = 1 . that it is possible to tune the current in the system fromone which is completely spin-polarized to an unpolarizedcharge current just with the help of an external gate.The origin for such transitions from semi-conducting tometallic in the gated ZGNRs and from a spin-polarizedto a spin-degenerate response in the Klein edged ZGNRscan be rationalized as follows. The ground state of bothsystems are entirely composed of magnetic edge states,which produce their respective band gaps. An exter-nal gate voltage does break the spin polarization of thesystem due to the change of the charge number at theedges. As a consequence, a controlled switch from aspin-splitting gap to metallicity is observed. One notesthat the application of an external magnetic field in cleanZGNR also yields a similar effect and giant magentore-sistance [17].In summary, we have studied the effect of gate volt-age on the band-structure and conductance profiles ofa ZGNR and a Klein edged ZGNR, based on the mean-field spin-unrestricted Hubbard model. The gate-injectedcharge density was obtained self-consistently, and con-ductance calculations were performed using the Lan-dauer’s formalism. We have found that an external gatecould tune the spin-induced band-gap in clean ZGNRs orin Klein edged ZGNR. In the former case, this leads to aswitching-on of the charge current, whereas in the lattercase, a transition from a pure spin current to a com-pletely unpolarized charge current is achieved. Thesefeatures could eventually help in designing GNR basedcharge and spin switches and FETs in future all-carboncircuits.Support from the Alexander von Humboldt Founda-tion and computing facilities provided by the ZIH at theDresden University of Technology are duly acknowledged.This work was partially supported by the EU projectCARDEQ under grant N0 IST-021285-2. C ondu c t a n ce ( e / h ) -1 0 1051015 (a)(b) gF E − E [eV]
FIG. 4: (color online) Conductance of the Klein edged ZGNRwith respect to the incident energy of the electrons ( E ) scaledby E gF for (a) V g = 0 (b) V g = 1 . et. al. , Science , 666 (2004); C. Berger et. al. , Science , 1191 (2006).[2] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S.Novoselov, A.K. Geim, Rev. Mod. Phys. , 109 (2009).[3] M.Y. Han, B. Ozyilmaz, Y. Zhang, Ph. Kim, Phys. Rev.Lett. , 206805 (2007).[4] M.C. Lemme et. al. , IEEE Electron Device Lett. , 282(2007); Z. Chen, Y.-M. Lin, M.J. Rooks, Ph. Avouris,Physica E , 228 (2007); X. Li et. al. , Science , 1229(2008); X. Wang et. al. , Phys. Rev. Lett. , 206803(2008).[5] K. Nakada et. al. , Phys. Rev. B , 17 954 (1996); K.Wakabayashi, Phys. Rev. B , 125428 (2001); K. Sasaki,S. Saito, R. Saito, Appl. Phys. Lett. , 113110 (2006).[6] L.G. Cancado et. al. , Phys. Rev. Lett. , 247401 (2004);C. Casiraghi, A. Hartschuh, H. Qian, S. Piscanec, C.Georgi, A. Fasoli, K.S: Novoselov, D.M. Basko, A.C. Fer-rari, Nano Letters , 1433 (2009); Y. Kobayashi, K.I.Fukui, T. Enoki, K. Kusakabe, Phys. Rev. B , 125415(2006); C.O. Girit et. al. , Science , 1705 (2009); J.C.Meyer, C. Kisielowski, R. Erni, M.D. Rossell, M.F. Crom-mie, A. Zettl, Nano Lett. (11), 3582 (2008); X. Jia et.al. , Science , 1701 (2009). [7] K. Wakabayashi, Carbon-based Magnetism-An overviewof the magnetism of metal free carbon-based compoundsand materials (Elsevier, Amsterdam, 2005), pp. 279-304.[8] D.J. Klein and L. Bytautas, J. Phys. Chem. A , 5196(1999)[9] Z. Liu, K. Suenaga, P.J.F. Harris, S. Iijima, Phys. Rev.Lett. , 015501 (2009);[10] C. Stampfer et. al. , Phys. Rev. Lett. , 056403 (2009);F. Molitor et. al. , Phys. Rev. B , 075426 (2009); K.A.Ritter J.W. Lyding, Nature Materials , 235 (2009).[11] D.A. Areshkin, D. Gunlycke, C.T. White, Nano Lett. , 204 (2007); D. Querlioz et al. , Appl. Phys. Lett. ,042108 (2008); M. Evaldsson, I.V. Zozoulenko, H. Xu,T. Heinzel, Phys. Rev. B , 161407(R) (2008); E.R.Mucciolo, A.H. Castro Neto, C.H. Lewenkopf, Phys. Rev.B , 075407 (2009); A. Cresti et al. , Nano Research ,361 (2008); A. Cresti and S. Roche, Phys. Rev. B ,233404 (2009).[12] Y.-W. Son, Marvin L. Cohen, and S.G. Louie, Nature , 349 (2006). See also Y.-W. Son, Marvin L. Cohen,and S.G. Louie, Phys. Rev. Lett. , 216803 (2006).[13] O.V. Yazyev, M.I. Katsnelson, Phys. Rev. Lett. ,047209 (2008); M. Wimmer, I. Adagideli, S. Berber,D. Tom´anek, K. Richter, Phys. Rev. Lett. , 177207(2008); B. Huang, F. Liu, J. Wu, B.-L. Gu, and W. Duan,Phys. Rev. B 77, , 155427 (2008); B. Huang, M. Liu, N. Su, J. Wu,W. Duan, B.-L. Gu, F. Liu, Phys. Rev. Lett. , 166404(2009); L. Brey and H. A. Fertig Phys. Rev. B , 205435(2007).[14] S. Dutta, A. K. Manna, and S. K. Pati, Phys. Rev. Lett. , 096601 (2009); S. Dutta and S.K. Pati, J. Phys.Chem. , 1333 (2008); O. Hod, V. Barone, J. E. Per-alta and G. E. Scuseria, Nano Lett., , 2295 (2007).[15] F. Munoz-Rojas, J. Fern´andez-Rossier, and J.J. Palacios,Phys. Rev. Lett. , 136810 (2009); F. Munoz-Rojas,D. Jacob, J. Fern´andez-Rossier, and J.J. Palacios, Phys.Rev. B , 195417 (2006); W. Y. Kim and K. S. Kim,Nature Nanotechnology , 408 (2008).[16] J. Fern´andez-Rossier, Phys. Rev. B , 075430 (2008).[17] J. Fern´andez-Rossier and J. J. Palacios, Phys. Rev. Lett. , 177204 (2007); J. J. Palacios, J. Fern´andez-Rossier,and L. Brey, Phys. Rev. B , 195428 (2008).D. A.Areshkin and B. K. Nikolic, Phys. Rev. B , 205430(2009).[18] E.H. Lieb, Phys. Rev. Lett. , 1201 (1989).[19] K. Kusakabe and M. Maruyama, Phys. Rev. B ,092406 (2003).[20] M. Inui, S.A. Trugman, and E. Abrahams, Phys. Rev. B49